Problem: Given that $8^{-1} \equiv 85 \pmod{97}$, find $64^{-1} \pmod{97}$, as a residue modulo 97.  (Give an answer between 0 and 96, inclusive.)
Solution: Since $8^{-1} \equiv 85 \pmod{97}$, $64^{-1} \equiv (8^2)^{-1} \equiv (8^{-1})^2 \equiv 85^2 \equiv \boxed{47} \pmod{97}$.